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0^0

askamathematician.com

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Re: 0^0

#31

Post by forrestthewoods »

quchen wrote: post #7

That's a rather long text to say "it's an arbitrary -- and conveniently chosen -- definition of a special case of the power function similar to how 1 is not prime". I also think the presentation was chosen poorly: lots of wrong information before the correct approach is presented. Wikipedia is probably a better source here: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...

I love Wikipedia. It's amazing. It makes the world a better place. I'm a pretty decent programmer. I do video games so I do lots of 3d math. I'd say I'm decent at that as well.

I hate Wikipedia for math. Absolutely hate it. Unless you are a mathematician by trade Wikipedia is damn near useless for learning new math concepts. I don't even bother checking it anymore.

Re: 0^0

#32

Post by ipince »

azinman2 wrote: post #22

Another good reminder on how math itself is arbitrary and made up by humans (often for what's simplest/easiest), and not handed down to us by God. Luckily it's an extremely useful and extendable made up system. I see this all the time with AI/machine learning. Most algorithms are based on assumptions that make the math work out better rather than being aligned with some "fundamental truth." The world is not linear, b…

I do think there is some "fundamental truth" (whether it was "handed down to us by God" or not. Sure, maybe our math system is not able to fully express that truth (and in fact, we are pretty sure it is incomplete--that is unable to proof certain truths), but that doesn't mean it's arbitrary..

Re: 0^0

#33

Post by elwell »

In any case, zero is a different animal than any other number. It can't actually exist in the physical world like other numbers; it's definition is non-existence. Zero is merely conceptual. Therefore, it can't easily fit into the picture of mathematics; at least not without requiring the definition of special cases.

Re: 0^0

#34

Post by jomtung »

ryannevius wrote: post #17

I understand the "math"...the numbers...the work on paper. But how does that translate to something useful in the real world? That, after all, is what useful math helps us do...solve problems for the real, tangible world. Saying that 0^0 = 1 is a cool math game; but translate 0 into something in the real world (i.e. nothing, none, etc.)...and trying to make something out of it other than 0 or "indeterminate" starts t…

From the article's example the simplicity of the binomial formula is extremely useful compared to a formula that would have to account for the case where k=0. Another commenter pointed out the useful elegance of 0log0=0 for physicists. These are the real world applications for mathematicians choosing definitions directly. Saying that an idea may be defined in many ways is correct, but choosing a working definition for the system helps to apply the definition to appropriate concepts. This is what mathematicians are doing when choosing a specific definition instead of saying that any definition will work.

One could argue the entire field of Real Analysis was formed because Calculus showed the world that we didn't really have those definitions, but they were needed. There are cases where the integral of the derivative does not equal the derivative of the integral (violation of the fundamental theorem of calculus) without having a specific epsilon-delta definition of limit.

Also, zero is not always the same as nothing or none. Zero is an abstract number that some have decided is useful to represent nothing or an empty set, but really comes from an abstract idea that you can count nothing and have a number. This goes back to the fact that numbers are pretty useful ideas regardless if you may consider one of them a function or not.

Re: 0^0

#35

Post by na85 »

quchen wrote: post #7

That's a rather long text to say "it's an arbitrary -- and conveniently chosen -- definition of a special case of the power function similar to how 1 is not prime". I also think the presentation was chosen poorly: lots of wrong information before the correct approach is presented. Wikipedia is probably a better source here: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...

My experience has usually been that Wikipedia is an abysmally bad tool for learning mathematics. The articles seem to be written by someone who has zero clue how to teach the concepts and merely is trying to wow the reader with their proof-writing skills.

Re: 0^0

#36

Post by Bahamut »

calcsam wrote: post #24

0 ^ any positive power = 0 0 ^ any negative power = 1/0 = undefined = +- inf So strictly only the right limit as n --> 0 of 0^n = 0. Not the limit .

0 is not defined as 0 * x = 0 - that is a derived formula. 0 is defined as the additive identity, i.e. x + 0 = x = 0 + x. It is a unique number.

0 * x = 0 is proven by noting that 0 * x = (0 + 0) * x = 0 * x + 0 * x (distributive property) and then subtracting the additive inverse of 0 * x from both sides to get that 0 = 0 * x.

However, this says nothing about 0^0, and one cannot talk about 0^(-n) for natural number n since 0 has no multiplicative inverse.

Also your argument on limits is not correct - you chose a particular path of approach for the expression y^x fixed along y = 0. Looking at another angle, the limit of x^x as x approaches 0 is clearly 1 as reasoned in the article, so this causes a clear disagreement here since you can argue for different values to make sense by tweaking the path of approach of the two dimensional function y^x appropriately.

Re: 0^0

#37

Post by dllthomas »

baddox wrote: post #8
mcweaksauce wrote: post #2

This is ridiculous. Doesn't this seem counter-intuitive for it to be anything else besides 0 or undefined?

If we don't have a specific mathematical context, then saying it's undefined is intuitive to me. Without context, 0 is no more intuitive to me than 1. These two statements are equally intuitive to me, but they give different results for 0^0: "Zero raised to any power is still just zero." "Any number raised to the zeroth power is one."

For me, intuition-wise, I'd order it "undefined, 1, 0".

There are 3 cases for 1 and one case for 0 that immediately spring to my mind when considering the problem:

    0) Limit of 0^x, as x approaches 0 (from above).
    1a) Limit of x^0 as x approaches 0 (from either direction).
    1b) Limit of x^x as x approaches 0 (from above).
    1c) "What did you multiply by 3 once, to get 3^1?  So, multiplying 1 by zero, zero times..."
Limits here, simplified to intuition level, being "what would you need to fill that hole in the graph?" The fact that these disagree would be why I'd assume undefined, but the case for 1 seems stronger (to me).

Re: 0^0

#38

Post by na85 »

kazagistar wrote: post #20
Bahamut wrote: post #15

Earlier quoted context omitted.

Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.

Sort of like how we use terse symbols for everything because ink and papyrus is expensive.

And because if you're Isaac Newton and you're writing pages and pages of proofs, you naturally want to use a shorthand in order to save yourself time, energy, and from repetitive strain injury.

Re: 0^0

#39

Post by jomtung »

dllthomas wrote: post #28
ryannevius wrote: post #17

I understand the "math"...the numbers...the work on paper. But how does that translate to something useful in the real world? That, after all, is what useful math helps us do...solve problems for the real, tangible world. Saying that 0^0 = 1 is a cool math game; but translate 0 into something in the real world (i.e. nothing, none, etc.)...and trying to make something out of it other than 0 or "indeterminate" starts t…

What is "^ 0" in general, that it "makes" 1 "out of" 2, 3, 4, etc?

using the definition in the article where X^n = 1 * X * X ...

x^2 = 1XX x^1 = 1*X x^0 = 1

Re: 0^0

#40

Post by lovamova »

dinkumthinkum wrote: post #27
azinman2 wrote: post #22

Another good reminder on how math itself is arbitrary and made up by humans (often for what's simplest/easiest), and not handed down to us by God. Luckily it's an extremely useful and extendable made up system. I see this all the time with AI/machine learning. Most algorithms are based on assumptions that make the math work out better rather than being aligned with some "fundamental truth." The world is not linear, b…

I don't think math is arbitrary at all.

It is. We invented the arabic numerals because they were easy to draw and we could written any numbers with them. Just like we invented higher lever computer languages instead of using assembly.

See what Fibonacci used to say in his first book Liber Abaci about using arabic numerals. http://en.wikipedia.org/wiki/Liber_Abaci

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